Applied Analysis REU
Research Experiences for Undergraduates
May 31 - July 25, 2026
Research Experiences for Undergraduates
May 31 - July 25, 2026
All the carefully designed projects are in the area of applied mathematics, emphasizing the applications of the mathematical tools that an undergraduate student can master.
Projects are computational, theoretical, or a combination of these. The required techniques are mostly covered in calculus and differential equations undergraduate courses. Weeks one and two consist of intensive courses on dynamical systems, one-dimensional conservation laws, basic measure theory, and one-dimensional optimal transport.
Possible Topics Offered
Faculty Mentor: Charis Tsikkou
Conservation laws and balance laws arise in a wide range of applications, including gas dynamics, fluid mechanics, aerodynamics, geophysical flows, biological transport, and traffic modeling. These systems describe the evolution of quantities such as mass, momentum, and energy and are typically formulated as hyperbolic systems of partial differential equations. A fundamental difficulty in their analysis is the formation of discontinuities (shock waves) in finite time, even from smooth initial data, which necessitates the study of weak solutions.
A central building block in the theory is the Riemann problem, an initial-value problem with piecewise-constant data separated by a single discontinuity. While classical Riemann solutions consist of shocks, rarefaction waves, and contact discontinuities, certain nonlinear systems admit singular shocks, in which one or more state variables develop extreme concentration in the form of weighted Dirac delta measures. Understanding when such solutions arise and how they should be interpreted is a central theme of this research topic.
In Summer 2024, the focus was on a system of two balance laws of Keyfitz–Kranzer type with a time-dependent generalized Chaplygin gas pressure. This model exhibits negative pressure and is motivated by applications in cosmology, where Chaplygin gas models are used to describe early-universe dynamics and late-time expansion. In [5], the primary objective was to analyze the structure of non-self-similar Riemann solutions, including singular shocks. Due to the explicit time dependence of the model, classical self-similarity breaks down, leading to Riemann solutions that change structure across different time intervals. We identified regimes in which classical and singular solutions coexist and demonstrated how transitions between these regimes occur. The analytical findings were validated numerically using the local Lax–Friedrichs scheme.
In Summer 2025, we studied a system consisting of one conservation law and one Keyfitz–Kranzer-type balance law, with a time-dependent source term. Despite its minimal structure, the system captures essential features of transport under density constraints and serves as a prototype for models in biological aggregation, granular flow, and traffic congestion. Our approach was two-fold. In [3], we assumed that the source term is zero and provided a detailed classification of self-similar Riemann solutions, including both classical wave patterns and non-classical overcompressive delta shocks. In particular, we showed that these singular shocks arise as limits of smooth solutions under Dafermos regularization. The analysis relied on blow-up techniques within the framework of Geometric Singular Perturbation Theory (GSPT), which allowed us to resolve the internal structure of the singular solutions.
In [4], we kept the time-dependent source term, which led to a richer solution landscape and a further breakdown of self-similarity. We demonstrated that the associated Riemann problem admits non-self-similar solutions involving overcompressive delta shocks, vacuum states, and transitions across a critical density threshold where mobility vanishes and characteristic speeds degenerate.
In all three projects, we confirmed that the solutions we constructed satisfy the governing equations in the sense of distributions. In the last two projects, we additionally used both direct substitution into the weak formulation and the shadow wave method. While these results establish the existence and internal structure of singular shock solutions, they also highlight several fundamental open questions that remain largely unresolved. A central issue is uniqueness. For systems admitting singular shocks, weak solutions are typically non-unique, and standard entropy conditions are often insufficient to select a physically relevant solution. This raises the question of whether singular shocks can be uniquely characterized by additional admissibility criteria, such as kinetic relations, viscous regularizations, or vanishing viscosity limits. Understanding which regularizations lead to which singular solutions is essential for both theory and computation.
From a modeling perspective, understanding the physical meaning of singular shocks is very important. In applications like crowd dynamics, granular flow, or compressible media with density limits, singular solutions may correspond to extreme concentration effects, jamming phenomena, or phase transitions. Establishing a clearer link between mathematical singularities and observable physical behavior is a crucial step toward validating and interpreting these models.
Additional directions for future research include:
Faculty Mentor: Adrian Tudorascu
In 2023 R. Hynd and I studied the long-time asymptotic behavior of the Sticky Particles dynamics on the real line. While the time average of the Sticky Particles Lagrangian map has a limit which arises as a general property of projections onto closed convex cones, we proved that the map itself has an asymptotic limit in the case where the Sticky Particles dynamics is constrained to a compact set. R. Hynd and I left open the question of whether there was a specific rate of decay. In [6], we proved that there is no such thing, i.e. sticky particles solutions can converge to equilibrium arbitrarily slowly.
In order to extend the above queries to problems with interaction potentials, also in the summer of 2024 our REU group performed a comprehensive study of the sticky particles solutions to the one-dimensional pressureless attractive Euler-Poisson system. We first provided a Lagrangian map characterization of the sticky particles solutions as projections onto the convex cone of square-integrable, essentially nondecreasing functions by following closely the approach employed earlier by Natile & Savare for the pressureless Euler case. The asymptotic behavior of the sticky particles solutions was the second main objective of that work; we obtained explicit exact collapse times into the equilibrium whenever such collapse occurs. In general, we proved in [1] that the sticky particles solution converges to equilibrium in the 1-Wasserstein distance at an explicit rate.
In the summer of 2025, the main objective of the REU group was a study of the asymptotic behavior of distributional solutions to the one-dimensional repulsive pressureless Euler-Poisson system. The system is a model for the dynamics of a mass distribution evolving on ℝ whose masses exert outward forces on one another. A discrete (describing the evolution of finitely many particles) solution is called sticky if, upon collision, particles stick together and move as one for all subsequent time, according to the conservation of mass and momentum principles. In [2], we proved results on the total energy (Hamiltonian) of the system and demonstrated the existence and uniqueness of so-called "perfect" states, where the Hamiltonian is constant over all time and the solution converges to equilibrium, a single stationary particle. We provided a necessary and a sufficient condition for finite-time collapse, and presented a quadratic envelope within which a solution must remain in order to collapse. We demonstrated various (counter)examples that illustrate the unique behavior of the repulsive scheme with the sticky condition, analytically and with a computer simulation.
Note, additional projects may be added to this list. If you have a special request, please contact the directors for help.
[1] B. Becerra, J. Linderoth, H. Pesin, A. Tudorascu, and R. Wassink, “Sticky Particles solutions for the attractive pressureless Euler-Poisson system; a projection formula and asymptotic behavior,” Journal of Mathematical Analysis and Applications, 550 (2), 2025, 129553.
https://doi.org/10.1016/j.jmaa.2025.129553
[2] N. Biglin, J. Crachiola, J. Curtis, T. Kunz, O. Maralappanavar, A. Tudorascu, “On the asymptotic behavior of the Repulsive Pressureless Euler-Poisson System.” Under Review
[3] J. Culver, A. Ayres, E. Halloran, R. Lin, E. Peng, and C. Tsikkou, “An Analysis of the Riemann Problem for a $2\times 2$ System of Keyfitz-Kranzer Type Conservation Laws Using Shadow Waves and Dafermos Regularization,” arXiv preprint arXiv:2508.05927. Under Review
[4] J. Culver, A. Ayres, E. Halloran, R. Lin, E. Peng, and C. Tsikkou, “An Analysis of the Riemann Problem for a $2\times 2$ System of Keyfitz-Kranzer Type Balance Laws With a Time-Dependent Source Term,” Physics of Fluids, 37 (11), 2025, 113340.
https://doi.org/10.1063/5.0296696
[5] J. Frew, N. Keyser, E. Kim, G. Paddock, C. Toumbleston, S. Wilson, and C. Tsikkou, “An analysis of a 2 × 2 Keyfitz–Kranzer type balance system with varying generalized Chaplygin gas,” Physics of Fluids, 36 (9), 2024, 096132. https://doi.org/10.1063/5.0231413
[6] A. Tudorascu, and R. Wassink, “Compactly supported Sticky Particles solutions decay to equilibrium arbitrarily slowly”, to appear in Methods and Applications of Analysis.